Inverse of Piecewise-Defined function

Question

I'm stuck on this problem:

Let $f$ be defined by f(x){ \begin{array}{cl} 2-x & \text{ if } x \leq 1, \\ 2x-x^2 & \text{ if } x>1. \end{array} Calculate $f^{-1}(-3)+f^{-1}(0)+f^{-1}(3)$.

It's difficult because I've never dealt with inverses in piecewise defined functions! I tried doing the inverses separately, but it's not working, as I get an imaginary number for $f^{-1}(3)$. Can somebody provide a solution that is easily understandable for somebody like me? Thanks!

Answer 1

hint

To find $ f^{-1}(Y) $, you must solve the two equations

$$2-x=Y \text{ with } x\le 1$$ and $$2x-x^2=Y \text{ with } x>1$$

this gives $ f^{-1}(3)=-1$.

You should find that $$f^{-1}(0)=2 \text{ and } f^{-1}(-3)=3$$